2020 AMC 8 Problem 20

Question

A scientist walking through a forest recorded as integers the heights of

5

trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?

\begin{array}{|c|c|}\hline T_1&\underline{\phantom{00}}\,\mathrm m\\T_2&11\,\mathrm m\\T_3&\underline{\phantom{00}}\,\mathrm m\\T_4&\underline{\phantom{00}}\,\mathrm m\\T_5&\underline{\phantom{00}}\,\mathrm m\\\hline\overline T&\underline{\phantom{00}}.2\,\mathrm m\\\hline\end{array}

22.2

24.2

33.2

35.2

37.2

Answer: B

Concepts:casework mean

Difficulty rating: 1370

Small hint:

Integer heights force the neighbors of

11

to be

22

Big hint:

The average ending in

0.2

means the total ends in

1

mod

5

Video solution:

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Written solution:

Tree

2

is

11

meters tall. Since all heights are integers and neighboring trees differ by a factor of

2,

trees

1

and

3

must both be

22

meters tall.

The first three trees total

22+11+22=55

meters. The possible integer pairs for trees

4

and

5

are

(11,22),

(44,22),

and

(44,88).

These give averages

17.6,

24.2,

and

37.4,

respectively. The notebook shows that the average ends in

0.2,

so it must be

24.2.

Thus, the correct answer is B.